Finite difference discretization of Elliptic equations: ID Problem ecture 2 and 3
✂✁☎✄✆✁✞✝✠✟☛✡☞✁✍✌✎✟✑✏✒✟✓✄✕✔✑✟☛✡☞✁☎✖✗✔✑✏✒✟✘✝✗✁☎✙✛✚✜✝✗✁✣✢✤✄ ✢✦✥★✧✪✩☎✩☎✁☎✫✬✝✠✁✣✔✭✧✯✮✱✰✲✚✜✝✠✁✣✢✤✄✆✖✛✳✵✴✜✡ ✶✷✏✒✢✤✸✕✩✣✟✓✹ ✺✻✟✑✔✼✝✠✰✕✏✒✟✓✖✾✽✿✚❀✄❂❁❄❃
1 Model problem 1.1 Poisson Equation in 1D Boundary Value Problem(BVP) (x)=∫(x) (0,1),u(0)=(1)=0,f Describes many simple physical phenomena(e.g) Deformation of an elastic bar Deformation of a string under tension Temperature distribution in a bar The Poisson equation in one dimension is in fact an ordinary differ tion. When dealing with ordinary differential equations we Poisson equation will be used here to illastrate numerical techniques for elliptic PDE's in multi-dimensions. Other techniques specialized for ordinary differen tial equations could be used if we were only interested in the one dimension Note 1 Poisson equation The Poisson equation(in R)is elliptic, per our classification. It is also coercive, nd symmetric(these concepts will be defined more precisely in the Finite Element lectures). These attributes are very important as regards umerical treatment. These properties are reflected in the fact(see first lecture) that the eigenvalues of -V-v are real and positive We denote by cm, more precisely, cm([0, 1]), the set of functions f(a): [0, 1]+ IR with continuous m derivat ives. Thus. cu denotes the set of continuous func Isly, Ck CC for k>
❅ ❆❇✬❈❊❉✛❋❍●❏■✼❇▲❑❊❋▼❉✛◆ ❖✼P◗❖ ❘❂❙❯❚◗❱❲❱❳❙❯❨✾❩❊❬❪❭✜❫✠❴❵❚❛❙❯❨❜❚◗❨❝❖✒❞ ❡❣❢❳❤❥✐❧❦✂♠ ♥▲♦❣♣✠q✗r✠s✉t✇✈✂①✤s❵②③♣✗④✎⑤✦t✇♦⑦⑥⑧②⑨④❲⑩❷❶❛♥▲①✆⑤❹❸ ❺❼❻❣❽❾❽❵❿➁➀➃➂❯➄✯➅⑧❿➁➀❣➂ ➆➈➇ ➀❊➉➊❿③➋❵➌ ➇ ➂☎➌➍❻⑧❿⑨➋➎➂✘➄✪❻⑧❿ ➇ ➂✼➄✷➋❵➌➏➅✎➉➑➐➓➒ ➆→➔ ↔❀↕➛➙✞➜✣➝✞➞➠➟❣↕➛➙❪➡➤➢➦➥❲➧➑➙➨➞➩➡✱➫➯➭➠↕✤➫❵➲❲➧✉➙➨➞➩➜➛➢➦➭➃➫❵➲❵↕➛➥❵➳➵➡➸↕▼➥⑦➢ ➆❀➣ ❿ ↕➵➺ ➻➯➺ ➂☎➼ ➽ ↔❀↕✣➾➁➳➎➝➨➡➤➢➦➚➨➞➩➳➵➥✕➳➦➾✼➢➦➥✕↕▼➭➪➢➵➙◗➚➨➞➪➜❀➟⑦➢➦➝ ➆✛➶ ➽ ↔❀↕✣➾➁➳➎➝➨➡➤➢➦➚➨➞➩➳➵➥✕➳➦➾✼➢✱➙◗➚➨➝✞➞➩➥❵➻➸➹❵➥➯➘✉↕➛➝❪➚➨↕▼➥⑦➙◗➞➩➳➵➥ ➆→➴ ➽➬➷↕▼➡➸➫❣↕▼➝✍➢➮➚✞➹❵➝➨↕✤➘✉➞➪➙◗➚➨➝✞➞➠➟❵➹❵➚➨➞➩➳➵➥❂➞➠➥❊➢✱➟➯➢➵➝ ➆❀➱ ✃➃❐❵❒→❮❼❰➮Ï➪Ð✍Ð▼❰➮ÑÒ❒✍Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ❊Ï❥Ñ✎❰➦Ñ❣❒➈Ø➦Ï❥Ù➸❒▼Ñ❵Ð☎Ï⑨❰➮Ñ✎Ï➪Ð→Ï❥Ñ➈Ú✣Õ➎Û✣Ö✓Õ➦Ñ✎❰➮Ü➨Ø➦Ï❥Ñ❣Õ➦Ü☎Ý➑Ø➮ÏÞ✛❒▼Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮ß❧❒✍Ó✣Ô➯Õ➦à Ö×Ï⑨❰➮Ñ❣á❝â✬❐❵❒▼Ñ★Ø➎❒✍Õ➮ßãÏ❥Ñ❲ä❍å✼Ï❥Ö➁❐✯❰➮Ü➨Ø➦Ï❥Ñ❣Õ➮Ü☎ÝæØ➮ÏÞ✛❒▼Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮ß✦❒✞Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ➯Ð❊å❯❒✎å✼Ï❥ß❥ß✦Õ➮ßçÐ▼❰❍Ô❲Ð▼❒✲Ö➁❐❵❒ èêéÜ☎Ï❥Ù➤❒❳ë✛Ñ❣❰➦ÖìÕ➦Ö③Ï⑨❰➦Ñ✆Öí❰❹Ï❥Ñ❣Ø➮Ï⑨Û✍Õ➦Öì❒✦Ø➮ÏÞ❀❒✣Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮Ö×Ï⑨❰➮Ñ➃á➤✃❣❐✉Ô❲Ð✍î ❻❽❹ï ❻➃ð î ❻❽❾❽ ➄ñ❻➃ð ð î✼❒✣ÖíÛ➛á✱✃➃❐❵❒ ❮❼❰➦Ï➪Ð✞Ð▼❰➦Ñò❒✍Ó✣Ô➯Õ➦Ö③Ï⑨❰➦Ñ✲å✼Ï❥ß❥ß✠ó✍❒➈Ô❲Ð✣❒✍Ø❹❐➯❒✣Ü✞❒✤Öí❰➸Ï❥ß❥ßãÔ❲Ð☎Ö③Ü✞Õ➮Öí❒✤Ñ➯Ô✉Ù➤❒▼Ü☎Ï⑨Û✞Õ➦ß✒Öí❒✞Û✍❐❲Ñ⑦Ï⑨Ó✣Ô➯❒✣Ð✼Ú✣❰➮Ü❹❒▼ß❥ßãÏéÖ×Ï⑨Û ❮✼ô✦õ➸öÐ➈Ï❥ÑòÙ▲Ô✉ßãÖ×Ï❥à◗Ø➮Ï❥Ù➤❒▼Ñ❵Ð☎Ï⑨❰➮Ñ➯Ð✣á✬÷❪Ö❥❐➯❒✣Ü➈Öí❒✍Û✞❐✉Ñ➯Ï⑨Ó▼Ô➯❒☎Ð✤Ðé❒✍Û✣Ï⑨Õ➦ßãÏ➠ø❾❒✞Ø✦Ú✣❰➦Ü▲❰➮Ü✞Ø➮Ï❥Ñ❣Õ➦Ü☎Ý❂Ø➦ÏÞ✛❒▼Ü➨❒✣Ñ⑦à Ö×Ï⑨Õ➮ß✑❒✍Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ➯Ð✬Û✍❰➦Ô✉ß➠ØÒó✍❒➑Ô❲Ð▼❒✞Ø✎ÏÚ➤å❯❒➑å❯❒✣Ü✞❒✕❰➦Ñ➯ßãÝ✎Ï❥Ñ➯Öí❒✣Ü➨❒✣Ð☎Öì❒✍Ø✲Ï❥Ñ❍Ö➁❐❵❒❂❰➮Ñ➃❒✆Ø➦Ï❥Ù➸❒▼Ñ❵Ð☎Ï⑨❰➮Ñ➃Õ➮ß Û✍Õ✇Ð▼❒❾á ùòú➓û☎ü✎ý þ➈ú❲ÿ✁✂➵ú☎✄✯ü☎✆✞✝✠✟➓û◗ÿ❛ú☎✄ ➷➲❵↕☛✡⑧➳➵➞➪➙➨➙➨➳➵➥✤↕✌☞➎➹⑦➢➮➚➨➞➩➳➵➥ ❿ ➞➠➥✎✍✏✒✑ ➂ ➞➩➙✗↕▼➭➩➭➩➞➠➫✉➚✞➞➩➜✔✓➦➫❣↕▼➝✠➳➵➹❵➝⑧➜✣➭➪➢➵➙✞➙◗➞✖✕⑦➜➛➢➮➚➨➞➩➳➵➥❧➺✗✍ì➚✠➞➪➙✠➢➦➭➪➙◗➳✛➜▼➳❲↕▼➝✍➜✣➞✙✘➵↕✔✓ ➳➵➝❧➫⑦➳❳➙◗➞➠➚➨➞✙✘➵↕✑➘✉↕✚✕➯➥➯➞ê➚✞↕✔✓➦➢➦➥⑦➘✤➙◗➧❲➡➸➡➸↕✣➚➨➝✞➞➪➜ ❿ ➚✞➲❵↕➛➙➨↕✓➜▼➳➵➥➯➜▼↕▼➫✉➚✍➙✗✛❼➞➩➭➠➭❳➟⑦↕✑➘✉↕✚✕⑦➥❵↕➛➘✤➡➸➳➎➝➨↕✘➫➯➝➨↕❾➜✣➞➪➙◗↕➛➭➠➧ ➞➩➥✬➚➨➲➯↕✢✜✠➞➩➥❵➞ê➚✞↕✤✣✘➭➩↕▼➡➸↕▼➥❳➚✓➭➩↕➛➜✣➚➨➹❵➝✞↕➛➙ ➂ ➺ ➷➲❵↕❾➙◗↕✦➢➮➚➨➚➨➝✞➞➠➟➯➹✉➚➨↕❾➙✑➢➵➝➨↕✥✘➵↕➛➝➨➧✱➞➠➡➸➫❣➳➵➝➨➚✞➢➵➥➎➚❼➢➵➙✓➝✞↕▼➻➎➢➵➝✞➘➯➙ ➥❲➹❵➡➸↕▼➝✞➞➩➜➛➢➦➭❳➚➨➝✞↕➛➢➦➚➨➡➸↕▼➥❳➚❾➺ ➷➲❵↕❾➙◗↕❪➫❵➝✞➳➵➫❣↕▼➝➨➚➨➞➩↕➛➙✼➢➵➝➨↕✑➝✞↕✚✦➯↕❾➜☎➚➨↕❾➘❹➞➩➥✱➚➨➲❵↕✑➾⑨➢➎➜☎➚ ❿ ➙➨↕▼↕☛✕➯➝✍➙◗➚⑧➭➩↕➛➜✣➚➨➹❵➝✞↕ ➂ ➚➨➲⑦➢➮➚❼➚➨➲➯↕✤↕▼➞➩➻➵↕▼➥☎✘➮➢➦➭➩➹❵↕❾➙✓➳➵➾ ❺★✧✑✪✩ ➢➦➝✞↕→➝✞↕➛➢➵➭✒➢➦➥➯➘✆➫❣➳➎➙➨➞ê➚✞➞✖✘➎↕➵➺ ùòú➓û☎ü✬✫ ➐✗✭ ✚✮✯✟✠✰➎ü✱ ✲ò↕✦➘❵↕▼➥❵➳➵➚➨↕✦➟❲➧ ➐✳✭ ✓✉➡➸➳➵➝✞↕❀➫➯➝➨↕❾➜✣➞➪➙◗↕➛➭➠➧✴✓ ➐✗✭✱❿✶✵➋❵➌ ➇✚✷ ➂ ✓❳➚➨➲❵↕✤➙➨↕✣➚❼➳➦➾✗➾➁➹❵➥➯➜✣➚➨➞➩➳➵➥➯➙ ➅⑧❿⑨➀❣➂❪➼✸✵➋❵➌ ✍✏✺✛❼➞➠➚➨➲❊➜▼➳➵➥❳➚➨➞➩➥❲➹❵➳➵➹➯➙☛✻ ➘❵↕▼➝✞➞✖✘➮➢➮➚✞➞✖✘➎↕➛➙➛➺ ➇✚✷✠✹ ➷➲❳➹⑦➙✪✓ ➐➓➒ ➘✉↕▼➥➯➳➦➚➨↕❾➙✑➚➨➲❵↕➈➙➨↕✣➚✜➳➵➾✠➜▼➳➵➥❳➚➨➞➩➥❲➹❵➳➵➹➯➙✑➾➁➹➯➥➯➜✽✼ ➚➨➞➩➳➵➥⑦➙▼➺✿✾❀➟☎✘❲➞➠➳➎➹➯➙◗➭➩➧✔✓ ➐❁❀❃❂❍➐✗✭ ➾➁➳➎➝★❄❆❅❇✻ò➺ ➇
Note 3 Green's function For this problem, the solution u can be written explicitly as (, y)f(y) where G(a, y) is the Green's function given by G(0)={01-) y o show this, we start by recalling that for any function which is twice differ entiable, there are constants CI C2, such that u()=G1+/u'(y)dy C If u satisfies the one dimensional Poisson equation, then C u(a)=C1+C2a f(z)dz dy f(a)dxdy Ly F(y)l F(y)dy F() yf(y)d g (a-y)f(y)dy, by proper attention to dummy variables. Finally, we obtain the general solution
❈❊❉✳❋✽●■❍ ❏▲❑✴●☎●✱▼❖◆P▲◗❙❘❚▼✯❯☎❋✶❱❲❉☎▼ ❳❩❨✴❬❪❭❴❫❩❵❜❛❞❝❩❬❡❨✔❢✸❣✖❤✂✐✬❥☎❭❡❫❩❤❦❛❴❨✔❣✙❧♠❭❴❵✙❨✔♥✬♦q♣✪r✔♥s❢✠❤✤t❞❬❴❵✖❭✶❭❡❤✪♥✉❤✚✈♠❝❩❣✙❵❜♣✚❵✖❭❴❣✙✇sr✔❛ ♦②①④③✠⑤✿⑥⑧⑦⑩⑨ ❶❸❷①❹③❁❺❴❻❩⑤✶❼❽①❹❻❩⑤✶❾✔❻ t❞❫❩❤✂❬❴❤ ❷ ①❹③✞❺✶❻❩⑤☛❵❜❛❪❭❴❫❩❤❃❿✢❬❡❤✪❤✂♥✗➀ ❛☛➁❹❧❩♥✸♣✽❭❡❵✖❨✴♥✉➂✔❵✙➃✔❤✪♥✬❢☎✇ ❷ ①❹③❁❺❴❻❩⑤✿⑥➅➄ ❻✳①❲➆❞➇❊③✠⑤ ➈❲❼ ➉▲➊➋❻s➊⑩③ ③②①❲➆❞➇❊❻♠⑤ ➈❲❼ ③■➊❇❻❆➊➌➆ ➍✞❨s❛❴❫❩❨➎t❸❭❴❫❩❵❜❛✂❥✳t➏❤➐❛❲❭❙r✱❬❴❭✥❢☎✇✉❬❴❤✌♣✪r✱❣✙❣✙❵✖♥❩➂❆❭❴❫✸r✱❭★➁❹❨✴❬❖r✔♥☎✇❆➁❹❧❩♥❚♣✽❭❴❵✙❨✔♥❊t❞❫❩❵❜♣❙❫q❵❜❛★❭❲t❞❵❜♣✚❤➐➑♠❵✖➒✠❤✂❬✶➓ ❤✪♥➔❭❡❵✙r✔❢❩❣✖❤✴❥☎❭❴❫❩❤✂❬❴❤❦r✔❬❴❤❖♣✚❨✴♥✸❛✶❭❡r✱♥➔❭❙❛✒→ ⑨ r✔♥✸➑✉→➏➣✴❥✸❛❴❧✸♣❙❫s❭❴❫✸r✱❭ ♦❽①❹③↔⑤↕⑥ → ⑨✯➙ ⑦➋➛ ❶ ♦✠➜➝①❹❻❩⑤❲❾✴❻ ♦➜ ①❹❻❩⑤➞⑥ →➣ ➙ ⑦➋➟ ❶ ♦➜ ➜ ①✁➠➔⑤✶❾✴➠❚➡ ➢➤➁❽♦q❛❴r✱❭❴❵❜❛❲➥✸❤✌❛☛❭❴❫❩❤✤❨✴♥❩❤✤➑♠❵✙✐➐❤✂♥✸❛❴❵✖❨✴♥✸r✱❣✗➦❽❨✴❵✙❛❡❛✶❨✴♥❆❤✌➧✴❧❚r➎❭❴❵✙❨✔♥❁❥☎❭❴❫❩❤✂♥ ♦➜ ①❹❻❩⑤✯⑥➌→➣ ➇ ⑦➋➟ ❶ ❼❽①✁➠➔⑤✶❾✴➠❚➡ ➍❪❫❩❤✂❬❴❤✪➁❹❨✔❬❡❤✔❥ ♦②①④③✠⑤✿⑥✺→ ⑨ ➙ →➣ ③➨➇ ⑦➋➛ ❶➫➩⑦➋➟ ❶ ❼❽①④➠☎⑤❲❾➔➠✔➭✉❾✴❻✠➡ ➯✥❤✚➥✸♥❩❵✙♥❩➂ ➲ ①❹❻❩⑤✯⑥⑧⑦➳➟ ❶ ❼❽①④➠☎⑤❲❾➔➠❚❺ t☛❤✢❨✴❢✸❛❴❤✪❬❡➃✔❤✥❭❴❫✸r✱❭ ⑦➛ ❶ ➩⑦➟ ❶ ❼❽①✁➠➔⑤✶❾✴➠✴➭■❾✔❻ ⑥ ⑦➛ ❶ ➲ ①❹❻❩⑤✶❾✔❻ ⑥ ➵❻➲ ①❹❻❩⑤➤➸ ➛❶ ➇ ⑦➳➛ ❶ ❻➲ ➜➝①❹❻❩⑤❲❾✴❻ ⑥ ③➲ ①❹③✠⑤✯➇ ⑦➋➛ ❶ ❻❩❼❽①❹❻❩⑤✶❾✔❻ ⑥ ⑦➛ ❶ ①④③➨➇➺❻❩⑤❴❼❽①❹❻❩⑤❲❾✴❻↔❺ ❢☎✇❦❝❩❬❡❨✔❝✠❤✪❬✯r✱❭✶❭❴❤✂♥➔❭❴❵✙❨✔♥➐❭❴❨❦➑♠❧✸✐➐✐✎✇❃➃➎r✱❬❡❵❜r✱❢❩❣✙❤✂❛✂➻✞❳②❵✙♥✸r✱❣✙❣✖✇✴❥✔t☛❤❞❨✔❢♠❭❙r✱❵✙♥➐❭❴❫❩❤❞➂✴❤✪♥❩❤✂❬❡r✔❣❩❛✶❨✴❣✖❧♠❭❡❵✖❨✴♥ ❵✙♥s❭❡❫❩❤❖➁❹❨✔❬❡✐ ➼
(a-uf(y)dy For our particular problem we can now impose the boundary conditions u(0) a(1)=0 to determine the constants CI and C2. Thus, after some arithmetic, u(a) (1-x)f(y)y+/x(1-y)f(y)d u(a)=G(a, g)f(y)dy We note that G(a, y) has the following properties IS ·G(x,y)≥0 for all z,y∈(0,1 G is a piecewise linear function of a for fixed y and vice versa The particular form of expressing the solution, in terms of the green function will be revisited when we address the topic of integral equation Note 4 Consider an elastic bar of unit length which is fixed at both ends and subjected to a tangential load per unit length p(a) p() u(a) da
➽❽➾❹➚✠➪➏➶➘➹❞➴✯➷⑩➹➏➬✂➚➨➮➳➱➳✃ ❐ ➾❹➚➨➮❊❒❩➪✶❮❽➾④❒♠➪✶❰✔❒↔Ï Ð❩Ñ✴Ò➏Ñ✴Ó❩Ò☛Ô✸Õ✱Ò❴Ö❴×❜Ø✚Ó❩Ù❜Õ✱Ò➏Ô❩Ò❡Ñ✔Ú❩Ù✙Û✪Ü➅Ý☛Û✢Ø✂Õ✱Þ➨Þ❩Ñ➎Ýß×✙Ü▲Ô❚Ñ➔à✶Û✥Ö❴á✸Û✥Ú✠Ñ✔Ó❩Þ❚â❩Õ✱Ò❡ãäØ✚Ñ✔Þ❚â♠×åÖ❡×✖Ñ✴Þ✸à ➽②➾✁æ✴➪ç➶ ➽❽➾❲è✌➪ç➶➌æ Ö❴Ñ➨â♠Û✪Ö❴Û✪Ò❡Ü▲×✖Þ✸Û✥Ö❡á❩Û❦Ø✚Ñ✴Þ✸à✶Ö❡Õ✱Þ➔Ö❙à ➹❪➴ Õ✔Þ✸â ➹☛➬✱é✯êá☎Ó✸à✂ë✸Õ➎ìíÖ❡Û✪Ò✒à❴Ñ✔Ü▲Û✤Õ✱Ò❡×åÖ❡á❩Ü▲Û✚Ö❴×❜Ø✱ë ➽②➾④➚✠➪✿➶ ➱✃ ❐ ❒✳➾✶è❞➮➺➚↔➪✶❮❽➾❹❒❩➪✶❰✔❒❦➷ ➱➋î ✃ ➚✞➾✶è✒➮➺❒❩➪❴❮❽➾❹❒❩➪❲❰✴❒↔ï Ñ✔Ò ➽❽➾❹➚↔➪✯➶ ➱ ➴ ❐⑧ð➾④➚❁ï✶❒❩➪❴❮❽➾❹❒❩➪❲❰✴❒↔Ï ñ❊Û✤Þ❩Ñ✱Ö❡Û✥Ö❡á✸Õ➎Ö ð ➾④➚❁ï❴❒♠➪ á✸Õ✔à☛Ö❡á❩Û❖ì❹Ñ✔Ù✙Ù✖Ñ➎Ý❞×✙Þ❩ò➐Ô❩Ò❡Ñ✔Ô✠Û✪Ò❴Ö❴×✙Û✂à✂ó ô ð ×❜à✒Ø✚Ñ✴Þ✴Ö❡×✖Þ☎Ó❩Ñ✴Ó✸à✂ë ô ð ×❜à✒à✶ã☎Ü▲Ü▲Û✚Ö❡Ò❴×❜Ø✢Û é ò é ð ➾❹➚✞ï✶❒❩➪✿➶ ð ➾❹❒↔ï✶➚↔➪ ë ô ð ➾④➚❁ï❴❒♠➪❞õ❇æ ì❹Ñ✔Ò✒Õ✔Ù✖Ù ➚✞ï✶❒✬ö❊➾✁æ❩ï✂è✌➪ ë ô ð ×❜à✒Õ➐Ô❩×✖Û✌Ø✚Û✂Ý❞×✙à❴Û❖Ù✖×✙Þ❩Û✂Õ✔Ò❪ì❹Ó❩Þ✸Ø✚Ö❴×✙Ñ✔Þ✉Ñ✱ì ➚ ì❹Ñ✔Ò❪÷❩ø♠Û✌â ❒ Õ✱Þ✸âsù☎×✙Ø✪Û✤ù✔Û✪Ò❙à❡Õ é êá❩Û✤Ô✸Õ✔Ò✶Ö❡×✙Ø✪Ó❩Ù❜Õ✱Ò❪ì❹Ñ✔Ò❡ÜúÑ✔ì②Û✪ø♠Ô❩Ò❴Û✌à❴à❴×✙Þ❩ò➐Ö❴á❩Û❦à❴Ñ✔Ù✙Ó♠Ö❴×✙Ñ✔Þ❁ë❩×✖Þ✉Ö❡Û✪Ò❡Üäà❞Ñ✱ì②Ö❴á❩Û➐û✢Ò❴Û✂Û✪Þsì❹Ó❩Þ✸Ø✽Ö❡×✖Ñ✴Þ✗ë Ý❞×✙Ù✖Ù✗Ú✠Û✤Ò❡Û✪ù☎×✙à❴×✖Ö❴Û✂âsÝ❞á❩Û✂ÞsÝ☛Û❦Õ✔â❩â❩Ò❴Û✌à❴à➏Ö❴á✸Û❖Ö❴Ñ✔Ô✸×✙Ø❖Ñ✔ì✞×✙Þ✴Ö❡Û✪ò✴Ò❡Õ✔Ù✠Û✌ü➔Ó✸Õ➎Ö❡×✖Ñ✴Þ✸à é ý❊þ✳ÿ✁✄✂ ☎✝✆✟✞✡✠✴ÿ☞☛✍✌✏✎✑✞✓✒ ✔Ñ✔Þ❚à✶×❜â♠Û✪Ò☛Õ✱ÞäÛ✂Ù✙Õ✴à❲Ö❡×✙Ø★Ú✸Õ✱Ò➏Ñ✱ì✗Ó❩Þ❩×✖Ö➏Ù✙Û✪Þ✸ò✱Ö❴á❆Ý❞á❩×✙Ø❙á❆×✙à✿÷❩ø♠Û✂â➨Õ✱Ö➏Ú✠Ñ✱Ö❡á➨Û✪Þ✸â✸à➏Õ✔Þ✸â➨à❴Ó❩Ú✡✕❲Û✂Ø✚Ö❴Û✌â Ö❴ÑäÕ✎Ö❡Õ✔Þ❩ò✔Û✂Þ➔Ö❴×❜Õ✱Ù✳Ù✙Ñ✴Õ✔âsÔ✠Û✪Ò✒Ó❩Þ✸×åÖ✒Ù✙Û✪Þ❩ò✔Ö❴á✗✖➾④➚✠➪✽é ✘
tively. From horizontal equilibrium we have gential displacement at a, respec- Let o(a) and u(a) be the axial stress and tan Aco -Ac(o +do Under the assumption of small displacements and a linearly elastic material we have where e is the modulus of elasticity and Ac is the area of the bar cross section Differentiating the constitutive equation and combining the two equations to eliminate o' we obtain the Poisson equation with f=p/(EAc) Note 5 String under transversal load Consider a string of unit length under tension T, which is subjected to a trans verse distributed load of magnitude p(a) per unit length p(a) da 6+d8 Let u(a) denote the transverse displacement at point a. Assuming small dis ements, so that the tension T can be taken as constant over the whole string and considering vertical equilibrium we have T(6+d6)-T6=pd The angle 8 can be related to the displacement u simply as dau Note the minus sign which is due to the fact that a positive u corresponds to a downwards displacement. Combining the two equations to eliminate the variable 8 we obtain the Poisson equation with f= p/T
✙✛✚✢✜✤✣✦✥★✧✑✩✤✪✬✫✮✭✰✯✱✥✲✧✑✩✴✳✑✚✵✜☞✶✓✚✷✪✹✸✡✺✻✪✬✼✾✽✿✜☞❀❁✚❂✽❁✽✴✪❃✫✮✭✰✜❁✪✬✫✮❄❃✚✢✫❅✜❁✺❆✪❃✼❇✭✓✺❆✽☞❈✓✼✻✪❃❉❊✚❂❋●✚✢✫❅✜✤✪✹✜❍✧❏■✓❀❁✚❂✽☞❈✑✚❂❉✁❑ ✜☞✺❆▲❃✚❂✼◆▼P❖✦◗✓❀❁❘❃❋❙✶✮❘❃❀❁✺◆❚❂❘❃✫❅✜❁✪❃✼✑✚❱❯❅❲✓✺◆✼❆✺❆✳✓❀☞✺❆❲✓❋❨❳✴✚✵✶✮✪❩▲❃✚ ❬✝❭ ✣✄❪ ❬❫❭ ✥✲✣✰❴❛❵P✣✛✩❝❜❡❞❢❵❃✧ ❣✤✫❤✭✡✚✢❀✴✜☞✶✮✚✵✪P✽☞✽☞❲✓❋●❈✡✜❁✺◆❘P✫✐❘✬❥❦✽✿❋❧✪✬✼❆✼❇✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❁✽♠✪❃✫✮✭✰✪♥✼◆✺❆✫✓✚❱✪✬❀❁✼◆▼●✚✢✼✻✪❃✽✿✜☞✺✻❉✝❋●✪✬✜☞✚❂❀☞✺✻✪✬✼✑❳♦✚ ✶✮✪❩▲P✚ ✣❢❜q♣ ✥★✯✗❴❛❵P✯❇✩❦❪r✯ ❵P✧ s ✥✲t✉❘✈❘❃✇❃✚P① ✽♠✼✻✪❩❳✉✩ ❳❍✶✓✚❂❀☞✚❫♣②✺✻✽③✜❁✶✓✚✝❋●❘✡✭✡❲✓✼❆❲✮✽✴❘✬❥④✚✢✼✻✪❃✽✿✜☞✺✻❉❊✺◆✜⑤▼●✪❃✫✮✭ ❬❫❭ ✺❆✽❝✜☞✶✮✚❫✪❃❀☞✚❱✪⑥❘✬❥✾✜❁✶✓✚❫✳✮✪✬❀♠❉❊❀❁❘P✽❁✽❝✽☞✚❂❉❊✜☞✺❆❘❃✫✛❖ ⑦✝✺⑨⑧❇✚✢❀❁✚✢✫❅✜☞✺✻✪✹✜❁✺◆✫✮❄❢✜☞✶✮✚✄❉✢❘❃✫✮✽✿✜☞✺◆✜☞❲✡✜❁✺◆▲P✚⑩✚❱❯❅❲✮✪✹✜❁✺◆❘P✫❛✪✬✫✮✭❛❉❊❘❃❋♥✳✓✺❆✫✓✺◆✫✮❄❢✜☞✶✮✚✰✜⑤❳✴❘❢✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✮✽❶✜☞❘ ✚✢✼❆✺❆❋✗✺❆✫✮✪✬✜☞✚✷✣✾❷✛❳✴✚✵❘❃✳✡✜❸✪✬✺❆✫✰✜❁✶✓✚⑥❹❦❘❃✺✻✽☞✽☞❘❃✫⑩✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✄❳❍✺⑨✜❁✶✏❺❻❜❼❞✾❽✡✥❾♣❬✝❭ ✩✁❖ ❿r➀✾➁✁➂❢➃ ➄❝➁✿➅❩➆❾➇❏➈➊➉❤➇③➋✱➂✬➅❢➁✿➅✹➌✡➇✱➍❅➎✡➂✬➅❩➍❱➌✡➏❍➏⑤➀✡➌❇➋ ➐❘❃✫❤✽✿✺✻✭✡✚✢❀♠✪✗✽⑤✜❁❀☞✺❆✫✓❄❶❘❃❥❏❲✓✫✓✺◆✜❍✼◆✚❂✫✓❄✬✜❁✶✰❲✓✫❤✭✡✚✢❀✴✜☞✚❂✫✮✽☞✺◆❘P✫✰➑⑥■✈❳❍✶✓✺✻❉❸✶✰✺✻✽♠✽✿❲✓✳✓➒⑤✚❂❉✁✜❁✚❂✭✐✜☞❘●✪⑥✜❁❀❁✪❃✫✮✽⑤❑ ▲❃✚❂❀❁✽☞✚✵✭✡✺❆✽✿✜☞❀❁✺❆✳✓❲✡✜☞✚❱✭✄✼◆❘❅✪❃✭⑩❘✬❥④❋❧✪❃❄❃✫✓✺◆✜☞❲✮✭✓✚❍❞④✥★✧❇✩✴❈❤✚❂❀✤❲✓✫✓✺◆✜✤✼❆✚✢✫✓❄❃✜☞✶✛❖ ✙✛✚✢✜⑥✯✱✥✲✧✑✩✷✭✓✚✢✫✓❘❃✜☞✚●✜☞✶✮✚❧✜☞❀❸✪✬✫✮✽☞▲❃✚❂❀❁✽☞✚●✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❶✪✹✜⑥❈✑❘❃✺❆✫❅✜❶✧❏❖✄➓✉✽❁✽✿❲✮❋✗✺❆✫✓❄❢✽☞❋❧✪✬✼❆✼❝✭✡✺✻✽⑤❑ ❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❁✽❂■✹✽✿❘✉✜☞✶✮✪✬✜❏✜❁✶✓✚❝✜❁✚✢✫✮✽☞✺❆❘❃✫⑥➑❛❉❂✪✬✫⑥✳✑✚③✜❸✪✬✇P✚✢✫❶✪P✽④❉✢❘❃✫✮✽✿✜❁✪❃✫P✜④❘✹▲❃✚✢❀❏✜☞✶✓✚✴❳❍✶✓❘P✼◆✚✴✽⑤✜❁❀☞✺❆✫✓❄✮■ ✪✬✫❤✭⑩❉✢❘❃✫✮✽☞✺❆✭✓✚✢❀❁✺◆✫✓❄✗▲❃✚❂❀✿✜❁✺❆❉❂✪✬✼✾✚❱❯P❲✮✺◆✼❆✺◆✳✮❀☞✺❆❲✓❋❙❳✴✚✵✶✮✪❩▲❃✚ ➑❶✥★➔✉❴❛❵❧➔P✩❦❪r➑✤➔⑥❜❡❞✏❵P✧❏→ ➣♠✶✓✚✷✪✬✫✮❄❃✼❆✚❫➔❧❉❂✪✬✫✄✳❤✚✷❀❁✚✢✼✻✪✹✜❁✚❂✭✰✜☞❘●✜☞✶✮✚⑥✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❍✯❢✽✿✺❆❋●❈✓✼❆▼⑩✪❃✽ ➔❶❜②❪ ❵❃✯ ❵❃✧ → ↔✤❘❃✜☞✚✏✜❁✶✓✚❢❋●✺❆✫❅❲❤✽⑩✽☞✺◆❄P✫↕❳❍✶✓✺❆❉❸✶➙✺✻✽✰✭✓❲✓✚❢✜❁❘❡✜☞✶✓✚❢❥✲✪❃❉✁✜✰✜☞✶✮✪✬✜✄✪❡❈✑❘P✽☞✺⑨✜❁✺◆▲P✚✏✯➛❉✢❘❃❀❁❀☞✚❱✽✿❈✑❘❃✫✮✭✮✽ ✜☞❘❢✪✏✭✓❘✹❳❍✫❅❳♠✪✬❀❸✭✓✽✵✭✡✺✻✽✿❈✮✼❆✪P❉❊✚✢❋●✚❂✫P✜❱❖ ➐❘❃❋♥✳✓✺❆✫✓✺◆✫✮❄❻✜☞✶✓✚●✜⑤❳✴❘✏✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✮✽❫✜❁❘✏✚✢✼❆✺◆❋●✺❆✫✮✪✹✜❁✚❧✜☞✶✓✚ ▲✹✪✬❀❁✺❆✪❃✳✓✼❆✚✝➔●❳✴✚❫❘P✳✡✜❁✪❃✺◆✫✄✜❁✶✓✚⑥❹❦❘❃✺✻✽☞✽☞❘❃✫✰✚❂❯❅❲✮✪✹✜❁✺◆❘P✫⑩❳❍✺◆✜☞✶❢❺❻❜❡❞✾❽❩➑⑥❖ ➜